Two-dimensional discrete-time laser model with chaos and bifurcations
We explore the local dynamical characteristics, chaos and bifurcations of a two-dimensional discrete laser model in $ \mathbb{R}_+^2 $. It is shown that for all $ a $, $ b $, $ c $ and $ p $, model has boundary fixed point $ P_{0y}(0, \frac{p}{c}) $, and the unique positive fixed point $ P^+_{xy}(\f...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-01-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023346?viewType=HTML |
| _version_ | 1811176119943036928 |
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| author | Abdul Qadeer Khan Mohammed Bakheet Almatrafi |
| author_facet | Abdul Qadeer Khan Mohammed Bakheet Almatrafi |
| author_sort | Abdul Qadeer Khan |
| collection | DOAJ |
| description | We explore the local dynamical characteristics, chaos and bifurcations of a two-dimensional discrete laser model in $ \mathbb{R}_+^2 $. It is shown that for all $ a $, $ b $, $ c $ and $ p $, model has boundary fixed point $ P_{0y}(0, \frac{p}{c}) $, and the unique positive fixed point $ P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a}) $ if $ p > \frac{b c}{a} $. Further, local dynamical characteristics with topological classifications for the fixed points $ P_{0y}(0, \frac{p}{c}) $ and $ P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a}) $ have explored by stability theory. It is investigated that flip bifurcation exists for the boundary fixed point $ P_{0y}(0, \frac{p}{c}) $, and also there exists a flip bifurcation if parameters vary in a small neighborhood of the unique positive fixed point $ P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a}) $. Moreover, it is also explored that for the fixed point $ P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a}) $, laser model undergoes a Neimark-Sacker bifurcation, and in the meantime stable invariant curve appears. Numerical simulations are implemented to verify not only obtain results but also exhibit complex dynamics of period $ -2 $, $ -3 $, $ -4 $, $ -5 $, $ -8 $ and $ -9 $. Further, maximum lyapunov exponents along with fractal dimension are computed numerically to validate chaotic behavior of the laser model. Lastly, feedback control method is utilized to stabilize chaos exists in the model. |
| first_indexed | 2024-04-10T19:46:58Z |
| format | Article |
| id | doaj.art-05662f4d374d42e995c5bc9030cb0ef9 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-10T19:46:58Z |
| publishDate | 2023-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-05662f4d374d42e995c5bc9030cb0ef92023-01-29T02:35:31ZengAIMS PressAIMS Mathematics2473-69882023-01-01836804682810.3934/math.2023346Two-dimensional discrete-time laser model with chaos and bifurcationsAbdul Qadeer Khan0Mohammed Bakheet Almatrafi11. Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan2. Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi ArabiaWe explore the local dynamical characteristics, chaos and bifurcations of a two-dimensional discrete laser model in $ \mathbb{R}_+^2 $. It is shown that for all $ a $, $ b $, $ c $ and $ p $, model has boundary fixed point $ P_{0y}(0, \frac{p}{c}) $, and the unique positive fixed point $ P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a}) $ if $ p > \frac{b c}{a} $. Further, local dynamical characteristics with topological classifications for the fixed points $ P_{0y}(0, \frac{p}{c}) $ and $ P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a}) $ have explored by stability theory. It is investigated that flip bifurcation exists for the boundary fixed point $ P_{0y}(0, \frac{p}{c}) $, and also there exists a flip bifurcation if parameters vary in a small neighborhood of the unique positive fixed point $ P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a}) $. Moreover, it is also explored that for the fixed point $ P^+_{xy}(\frac{ap-bc}{ab}, \frac{b}{a}) $, laser model undergoes a Neimark-Sacker bifurcation, and in the meantime stable invariant curve appears. Numerical simulations are implemented to verify not only obtain results but also exhibit complex dynamics of period $ -2 $, $ -3 $, $ -4 $, $ -5 $, $ -8 $ and $ -9 $. Further, maximum lyapunov exponents along with fractal dimension are computed numerically to validate chaotic behavior of the laser model. Lastly, feedback control method is utilized to stabilize chaos exists in the model.https://www.aimspress.com/article/doi/10.3934/math.2023346?viewType=HTMLlaser modelbifurcationchaos controlnumerical simulations |
| spellingShingle | Abdul Qadeer Khan Mohammed Bakheet Almatrafi Two-dimensional discrete-time laser model with chaos and bifurcations AIMS Mathematics laser model bifurcation chaos control numerical simulations |
| title | Two-dimensional discrete-time laser model with chaos and bifurcations |
| title_full | Two-dimensional discrete-time laser model with chaos and bifurcations |
| title_fullStr | Two-dimensional discrete-time laser model with chaos and bifurcations |
| title_full_unstemmed | Two-dimensional discrete-time laser model with chaos and bifurcations |
| title_short | Two-dimensional discrete-time laser model with chaos and bifurcations |
| title_sort | two dimensional discrete time laser model with chaos and bifurcations |
| topic | laser model bifurcation chaos control numerical simulations |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023346?viewType=HTML |
| work_keys_str_mv | AT abdulqadeerkhan twodimensionaldiscretetimelasermodelwithchaosandbifurcations AT mohammedbakheetalmatrafi twodimensionaldiscretetimelasermodelwithchaosandbifurcations |